Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

The commutation principle of Ramírez et al. (SIAM J Optim 23:687–694, 2013) proved in the setting Euclidean Jordan algebras says that when sum a real valued function h and spectral $$\Phi $$ is minimized/maximized over set E, any local optimizer at which Fréchet differentiable operator commutes with derivative $$h^{\prime }(a)$$ . In this note, we describe some analogs above result by assuming existence subgradient place (of h) obtaining strong commutativity relations. We show, for example: if solves problem $$\underset{E}{\max }\,(h+\Phi )$$ , then strongly every element subdifferential a; If E are convex $$\underset{E}{\min }\,h$$ negative a. These results improve known relations linear solutions variational inequality problems. establish these via geometric valid not only algebras, but also broader setting.